Equity, debt and moral hazard: The OptimalStructure of Banks’ Loss Absorbing Capacity∗

Misa Tanaka†and John Vourdas‡

May 16, 2016

Abstract

This paper develops a model to analyse the optimal ex-ante capital and totalloss absorbing capacity (TLAC) requirements, and the ex-post resolution of banks.Banks in our model are subject to two types of moral hazard: i) ex ante, theyhave the incentive to shirk on project monitoring in the presence of governmentguarantees on deposits, thus increasing the risk of a failure, and ii) ex post, poorlycapitalised, limitedly liable banks have the incentive to engage in asset dissipationby ’gambling for resurrection’. We demonstrate that ex-ante moral hazard canbe mitigated by ensuring sufficient TLAC plus equity capital buffer requirements,while prompt resolution of poorly capitalised banks can eliminate ex-post moralhazard. The optimal size of capital buffer and TLAC, and the optimal compositionof TLAC, depend on the social costs of bail in and bail out, as well as the probabilityof a system-wide shock. Our analysis suggests that, in response to an elevated riskof a system-wide shock, the regulator may want to increase capital buffers whilereducing TLAC requirements.

∗Disclaimer: Work in progress: preliminary and incomplete. Please do not quote without the permis-sion of the authors. The views expressed in this paper are those of the authors, and not necessarily thoseof the Bank of England, the Monetary Policy Committee (MPC), or the Financial Policy Committee(FPC). We thank Andrew Gimber, Elena Carletti and seminar participants at the Bank of England fortheir useful input.†Bank of England. Email: misa.tanaka@bankofengland.co.uk‡European University Institute, Florence. Email: john.vourdas@eui.eu

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1 Introduction

A key aim of the post-crisis global regulatory reform effort has been to end the problemof ‘too big to fail’. The ‘too big to fail’ problem arises if the government cannot crediblycommit not to bail out large, interconnected banks ex post, such that creditors underpricetheir credit risks ex ante. This in turn encourages banks’ shareholders and executives totake socially excessive risks, thus increasing the potential liabilities for taxpayers.

Post-crisis reform led to two new regulatory requirements for large, systemic banks.First, in order to reduce the likelihood of a failure of large, systemic banks, those banksthat the Financial Stability Board (FSB) classify as ‘global systemically important banks(G-SIBs)’ are now subject to additional capital buffer requirements on top of the Basel IIIrequirements and buffers. Unlike the minimum capital requirement, which banks have tomeet in all circumstances, capital buffers can be ‘used’ by banks in times of stress to absorblosses without entering into resolution. Second, in order to help ensure that systemicbanks can be resolved without requiring public support or threatening financial stability,G-SIBs will also be subject to a total loss absorbing capacity (TLAC) requirement from 1January 2019. The FSB’s Principles on Loss-absorbing and Recapitalisation Capacity ofG-SIBs in Resolution published in November 2015 suggests that TLAC can include bothregulatory capital instruments that are not used to meet regulatory capital buffers, aswell as other eligible unsecured debt instruments with residual maturity of over one year.The Principles suggests that the minimum TLAC must be at least 16% of the resolutiongroup’s risk-weighted assets from 1 January 2019 and at least 18% as from 1 January 2022.The Principles states that, while regulatory capital buffers must be ‘usable’ without entryinto resolution, a breach or likely breach of minimum TLAC requirement should be treatedas severely as a breach or likely breach of minimum capital requirements.

In this paper, we develop a simple framework which examines the optimal setting of thetrio of regulatory requirements: a minimum capital requirement and a minimum TLACrequirement, which need to be met at all times, and a capital buffer requirement, whichbanks can ‘use’ to absorb losses without entering resolution in stressed conditions. Inour model, banks are funded by equity, unsecured (bail-inable) debt and insured deposits.Following Allen, Carletti and Marquez (2015), we assume that credit and equity marketsare segmented, and that equity investors face a higher opportunity cost of investmentthan creditors. This makes equity funding not only privately but also socially morecostly relative to deposit and debt funding in our model. But we also assume thatequity has the advantage of being able to absorb losses ex post without generating socialcosts, while the imposition of losses on unsecured creditors (which we refer to as ‘bail-in’)and the deposit insurance fund which compensates the depositors (which we refer to as‘bail out’ as a short-hand) are both socially costly. The funding structure also affectsbanks’ incentive to engage in two types of moral hazard: i) ex ante, banks that maximiseshareholder returns have the incentive to shirk on costly project monitoring efforts inthe presence of (mispriced) government guarantees on deposits, thus increasing the riskof a failure, and ii) ex post, poorly capitalised, limitedly liable banks have the incentiveto engage in asset substitution by taking excessive risks at the expense of the depositinsurance fund (or taxpayers) and the long-term unsecured debt holders.

Our analysis yields the following key results. First, we demonstrate that ex-ante moralhazard can be mitigated by ensuring that banks’ total private loss absorbing capacity –i.e. TLAC plus equity capital buffer – is sufficiently high, but that the composition of thisprivate loss absorbing capacity is irrelevant for ex ante moral hazard when unsecured debt

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market can price risks and penalise risk-taking through higher interest rate. Second, weshow that, in order to prevent the ex post moral hazard, the resolution authority needsto step in and ‘bail in’ the unsecured debt as soon as the minimum capital requirement isbreached. Hence, the minimum capital requirement should be set at a level below whichshareholders’ ‘skin in the game’ is too low such that banks have the incentive to engage inasset substitution. Third, the optimal level of capital buffer is determined by a trade-offbetween the social cost of equity issuance and the benefit of reducing the probability andthe cost of a bank failure.

Our simple framework also enables an analysis of how changes in economic conditionsmight change the optimal trio of regulatory requirements. Our analysis suggests that ahigher bail-in cost calls for a higher capital buffer and a lower TLAC requirement, whereasa higher bail-out cost calls for both the capital buffer and the TLAC requirement to beraised. Furthermore, our analysis suggests that, in response to an elevated risk of asystem-wide shock, the regulator may want to increase the capital buffer while reducingthe TLAC requirement. This is because an increased risk of a system-wide shock increasesthe expected cost of bank insolvency, and an additional capital buffer helps to reduce therisk of bank insolvency.

The rest of the paper is organised as follows. Section 2 reviews the existing liter-ature. Section 3 develops the baseline model, in which the probability of a bad state,or a system-wide shock – which reduces asset return for all banks – is assumed to beexogenous: thus, we focus on the impact of ex-post moral hazard in the baseline analysis.Section 4 solves for the model equilibrium and clarifies the determinants of the optimalregulatory requirements. Section 5 endogenises the probability of a bad state, which isnow determined by banks’ decision about whether to monitor the project or not. Thisextended model allows us to analyse policies to mitigate ex-ante moral hazard. Section6 concludes.

2 Related literature

Our paper is related to a number of existing papers that have examined the impact ofcapital requirements on banks’ risk-taking incentives. In general, the conclusion fromthe existing literature on the impact of capital requirements on risk taking is ambiguous.For example, Hellman et al 2000 show that capital requirements have two counteractingeffects on the incentive for a bank to take on a more risky investment project. On the onehand, an increase in capital requirements increases shareholders’ ‘skin in the game’ byputting more of the shareholder’s equity capital at risk: this has the effect of reducing theincentive to take on risk. But on the other hand, an increase in capital requirements alsoreduces the bank’s profit margin, thus eroding its franchise value and incentivising it totake on greater risks. The authors conclude that the overall effect of capital requirementson banks’ risk-taking is ambiguous.

Acharya et al. (2013) analyse how capital requirements may address two moral hazardproblems in banking: asset substitution to socially inefficient risky loans, and managerialunder-provision of effort in loan monitoring. Optimal capital requirements trade off theneed to have banks sufficiently leveraged with unsecured debt that provide market disci-pline to incentivise bank managers to monitor their loans, versus the need to have banksbeing sufficiently financed by equity to avoid inefficient asset substitution. In a contextin which bank creditors anticipate a bailout, the optimal ex-ante regulation is in the formof a two-tiered capital requirement, consisting of a standard capital requirement and an

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additional special capital requirement which must be matched on the asset side of thebalance sheet with safe assets. The safe assets are unavailable to creditors upon failure ofa bank in order to ensure market discipline, and only available to shareholders to preventthem from shifting risk onto the bailout fund. This special capital account performs arole similar to the role of unsecured debt in our model.

Allen et al. (2011) consider how optimal capital regulation depends on the presence(or lack thereof) of deposit insurance, and competition in the deposit and credit markets.In their model, banks compete for deposits in a perfectly competitive deposit market,so they earn zero profits in expectation. In the absence of deposit insurance, marketdiscipline induces banks to fund themselves voluntarily with costly equity capital in orderto reduce the riskiness and corresponding required return on deposits. If deposits areinsured, the bank benefits from a source of funding which is insensitive to their choice ofrisk, leading them to fund themselves with too little capital from a societal perspective.The presence of insured deposits therefore gives rise to a role for capital requirements.

Repullo (2004) considers the effect of capital requirements in an imperfectly compet-itive banking industry. He shows that there are two equilibria: one prudent and onegambling equilibrium in which banks always invest in safe and risky assets respectively.Which equilibrium occurs depends on the nature of competition and capital requirements.If the market is moderately competitive introducing a flat capital requirement ensures theexistence of a prudent equilibrium as the franchise-value effect is muted, and the capital-at-risk effect incentivises the bank to act prudently.

The main contribution of our paper to this existing literature is to consider the opti-mal settings of the minimum capital requirement, TLAC requirement, and capital bufferin a single framework, instead of examining the determinants of the optimal capital re-quirement in isolation. There are, however, a few existing papers that have examined theissue of how contingent-convertible bonds (Cocos) and bail-in debt might affect banks’ in-centives. For example, Martynova and Perotti (2015) analyse how contingent-convertiblebonds affect banks’ risk-taking behaviour, and argue that Cocos that trigger when thebank is still solvent will reduce banks’ risk-shifting incentives more than bail-in bondswhich they assume will only convert into equity or written down when the bank becomesinsolvent. While we do not explicitly examine the role of Cocos, we also argue that,for bail-in bonds to reduce banks’ (ex post) risk-shifting incentives, they need to convertwhile equity value is still positive. Unlike Martynova and Perotti (2015), we also examinehow a bank’s ex ante risk-taking incentives are influenced by the funding mix and showthat equity and bail-in debt have the same disciplining effect on banks.

Our paper is most closely related to Mendicino, Nikolov and Suarez (2016), who alsoexamine the optimal size and composition of banks’ loss absorbing capacity. The keydifferences between our analysis and theirs is that Mendicino et al. (2016) assume that thebank maximises returns of insider equity holders plus private benefits, and that equityand bail-in debt are perfect substitutes as loss-absorbing instruments. In this set up,increasing the capital requirement reduces risk-shifting incentives but comes at the costof reducing the share of insider’s equity and increasing the incentive for private benefittaking. Our framework for pinning down the optimal requirements is different from theirs,as we focus mainly on the trade-off between the cost of equity funding and the benefit ofequity as a loss-absorbing instrument. We also distinguish minimum capital requirementand equity buffer more clearly than Mendicino et al (2016), as this distinction was a keypart of the post-crisis reform of the regulatory capital framework.

Our paper is also related to the literature on bank failure resolution. In our model,

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the rationale for resolution arises due to the need to intervene promptly to prevent un-dercapitalised banks from ‘gambling for resurrection’ by investing in excessively riskybut high return assets, as in Tanaka and Hoggarth (2006) and De Nicolo et al. (2014).The existing literature has also examined the possibility of time-inconsistent resolutionauthority (e.g. Mailath and Mester (1994)) and resolution authority/regulator that isimperfectly informed about banks’ projects (e.g. Vollmer and Wiese (2013)). In ouranalysis, however, we abstract from the possibilities of time-inconsistency and imperfectinformation, and assume that the resolution authority can intervene in the bank as soonas the minimum capital requirement is breached.

3 The model

This section outlines our baseline, three-period model. We assume that the regulatorinitially imposes regulatory requirements consisting of a minimum capital requirement,a TLAC requirement, and a capital buffer on banks that are ex ante identical. Thereare two states of the world, H (’good’) and L (’bad’), which determine the interim returnthat banks receive. Banks are ex post heterogeneous, because in the bad state, theyreceive different returns, which are unknown ex ante. In the baseline model, banksare only subject to ex post moral hazard: banks with low returns and high debt havethe incentives to engage in asset substitution – or gamble for resurrection – in order tomaximise shareholder returns at the expense of the unsecured debt holders and the depositinsurance fund (DIF). The presence of this ex post moral hazard creates a case for theresolution authority to intervene and resolve undercapitalised banks in order to preventthem from failing with larger losses – which also imply larger social costs – at a later date.

3.1 The set up

There are three periods: t = 0, 1, 2. All agents are risk-neutral, and the banking sectorconsists of a continuum of banks of mass 1. At t = 0, ex ante identical banks each have aninvestment opportunity which requires a unit of funds. Banks can fund this investmentopportunity through issuance of equity E0, unsecured, TLAC eligible (bail-inable) debt,G, and insured deposits, D. The balance sheet identity at t = 0 is given by:

E0 +G+D = 1

We define θ ≡ E0 + G, where θ is the uninsured debt and equity, and the share ofequity within this as e0. Thus, the liability side of the balance sheet at t = 0 can beredefined as E0 = θe0, G = θ(1− e0), and D = 1− θ. All debt holders are paid at t = 2if the bank remains in business. Insured deposits have a unit gross return in all states ofthe world because the deposit insurance fund (DIF) will cover any shortfall if the bankhas insufficient resources to pay depositors.1 Unsecured debt in this model can be bailedin when necessary as explained below, and carries interest rate i which fully prices inthe credit risk. When the bank is solvent, equity holders receive the residual value onceinsured depositors and unsecured debt holders have been paid; but when it is insolvent,then they are wiped out.

1Insured deposits can be also interpreted as any debt liabilities that are expected to be bailed out bythe government in the event of a bank failure.

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3.2 Ex ante regulatory requirements and investment (t=0)

At the start of t = 0, the regulator sets the regulatory requirements for ex ante identicalbanks. Specifically, the regulator sets θ and e0 which jointly determine E0, which is thecapital ratio that the bank is expected to maintain at t = 0, and G, which is the unsecureddebt ratio. In addition, the regulator also sets E∗, which is the minimum capital ratiorequirement that the bank has to maintain at t = 1 in order to remain in business. Thus,the difference between the initial capital ratio and the minimum capital ratio E0 − E∗

can be interpreted as the regulatory capital buffer, which can be used to absorb losseswithout the bank facing resolution when the return on assets at t = 1 turns out to below; and E∗ +G can be interpreted as the minimum TLAC requirement which the bankneeds satisfy at all times. Thus, θ represents the minimum TLAC plus the regulatorycapital buffer. We assume that the regulator chooses the minimum capital requirement(E∗), the capital buffer (E0 − E∗ = θe0 − E∗) and TLAC (E∗ + G = E∗ + θ(1 − e0))in order to maximise social welfare, whereas banks choose their investments and fundingstructure in order to maximise the returns to their shareholders.

We assume that, when setting these regulatory requirements, the regulator knows thatthe bank can invest in a project that yields return equal to RH in the ’good’ macroeco-nomic state which occurs with probability q and RL in the ’bad’ state which occurs withprobability 1− q at t = 1: 1− q could be interpreted as the probability of a system widestress which reduces return on assets across the banking sector. In this section we assumethat q is exogenous, but in Section 5 we endogenise the choice of q.

We assume that, in the high state when R = RH , banks will be solvent with certainty:RH > (1 − θ) + iθ(1 − e0), where i is the interest rate on unsecured debt. However,at the start of t = 0 the low return is stochastic such that RL ∼ Unif [0, Rmax], whereRmax ≤ RH . Thus, the solvency of each bank in the bad state depends on the realisationof RL relative to the level of equity E0. We interpret RL as bank ‘type’ which is ex anteuncertain, and the regulator sets all requirements without observing the realised RL forindividual banks. This captures the fact that the regulator sets the requirement for thebanking sector as a whole without necessarily knowing all the possible return outcomesunder a range of stress scenarios for each individual bank.

At the end of t = 0, RL is realised for each bank and becomes publicly observable:this makes banks heterogeneous ex post. The bank then issues deposits, uninsured debtand equity. We assume that all of the depositors, unsecured creditors and equity holdersare risk neutral and have access to a safe asset which yields a certain return normalised toequal 1. Following Allen, Carletti and Marquez (2015), we also assume that equity andcredit markets are segmented, and that, unlike depositors and creditors, equity investorscan directly participate in financial markets and therefore have access to an additionaloutside investment opportunity that yields a higher expected return, δ > 1. Thus, whileinsured depositors and unsecured creditors face an opportunity cost equal to 1 in lendingto the bank, equity investors face a private opportunity cost of δ > 1 in investing inbank equity. The assumptions of insured deposits (where the insurance premium is notfairly priced) and segmented markets create a departure from the Modigliani and Miller’s(1958) result that, in a frictionless world, the funding structure of a firm is irrelevant forits cost of funding. In our model, insured deposits are the least costly form of funding asthey benefit from the implicit subsidy provided by the deposit insurance. The unsecureddebt is the second most expensive form of funding, while equity is the most expensive,as equity investors face a higher private opportunity cost than creditors. Hence, banks’capital and TLAC regulations are binding in our model: in equilibrium, banks only issue

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as much equity and unsecured debt as required by the regulator, in order to maximise thebenefit of the subsidy from the deposit insurance scheme.

3.3 Resolution and bail in in the interim period (t=1)

At time t = 1, the macroeconomic state is realised. Each bank receives the interimreturn, which is either RH or RL depending on the macro state. If the interim returnis sufficiently low, such that a bank does not meet the pre-specified minimum regulatorycapital requirement E∗, the regulator (or the resolution authority) intervenes in the bankby converting unsecured debt into equity in order to recapitalise the bank.2 Specifically,the ‘bail in’ of unsecured debt occurs if the following condition holds:

RL −D − iGRL

=RL − (1− θ)− iθ(1− e0)

RL

< E∗ (1)

We assume that, in the absence of any regulatory intervention at t = 1, the bank canchoose between reinvesting the interim return either in a risky or a safe asset. The riskyasset yields a gross return γ > 1 at t = 2 with probability p and 0 with probability 1− p,whereas the safe asset is a simple storage technology which yields a unit gross return att = 2. We assume that the risky asset has a negative net present value, such that pγ < 1.Thus, the socially optimal choice is for the bank to reinvest its asset return at t = 1 inthe safe asset. However, the presence of long-term debt which only needs to be repaid att = 2, and (non actuarially fairly priced) deposit insurance both create the incentive forthe bank’s shareholders to ‘risk shift’, i.e. to take excessive risks at the expense of debtholders so as to maximise shareholder returns. As we show below, undercapitalised bankshave the incentive to ‘gamble for resurrection’ by investing in risky assets at t = 1, thusmagnifying the expected losses for creditors and the DIF. Thus, the minimum capitalrequirement E∗ needs to be set in such a way to allow the regulator to intervene to preventthis privately optimal but socially sub-optimal gambling behaviour, once this thresholdis breached. Our model does not provide a justification for having a separate trigger fora regulatory intervention – which we refer to as ‘resolution’ – based on the breach of theTLAC requirement.

3.4 Debt repayment in the final period (t=2)

At t = 2, all debt is repaid if the bank is solvent, and the equity holders receive theresidual return. If the bank is insolvent, insured depositors are paid in full by the DIFbut the unsecured debt holders receive zero. If the bank is insolvent but the value ofits assets exceed the liability to insured depositors, unsecured debt holders receive theresidual return after the payment to the depositors have been made; and equity holdersreceive zero return.

We assume that, while equity is expensive for banks (and as we explain later, forsociety) to issue, equity holders can absorb losses without imposing costs on the rest ofthe society. By contrast, we allow for the possibility that losses imposed on unsecureddebt holders via ‘bail in’ at t = 1, or at insolvency at t = 2, could give rise to a social costwhich depends on the size of the losses that are imposed on unsecured debt holders. We

2We assume that the resolution authority is able to credibly precommit to this triggger for intervention.Thus we are not focusing on the question addressed by Mailath and Mester about under which conditionsa resolution strategy is time inconsistent.

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allow for this possibility because unlikely equity, debt will need to be rolled over. Forexample, Avgouleas and Goodhart (2014) and Admati and Hellwig (2016) have raisedthe possibility that the imposition of losses on unsecured debt holder in one bank couldcause a dry-up of the market for such unsecured bank debt, especially when the financialsystem is already under stress. Furthermore, we also allow for the possibility that the‘bail out’ of depositors might give rise to a social cost which depends on the size of lossesimposed on the DIF: this captures the possibility that any DIF deficit will need to befunded by distortionary taxes which create a deadweight loss. The details of the modelspecification are described in Section 4.4, which examines how the regulator should setthe various requirements in order to maximise the social benefits.

The timing of the model is illustrated in Figure 1.

Figure 1: Timing of the model

4 Socially optimal regulatory requirements and res-

olution

We now solve this model backwards in order to illustrate the determinants of the sociallyoptimal minimum capital requirement, regulatory capital buffer and TLAC requirement.We demonstrate that ex ante regulatory requirements determine the share of insolventbanks in the bad state, as well as the size of the losses imposed on different stakeholdersof the bank – the shareholders, the unsecured debt holders and the deposit insurance fund

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(DIF). Hence, the regulatory requirements will need to be set by taking into account thepossible social costs associated with imposing losses on different stakeholders.

4.1 Optimal minimum capital requirement

At t = 1, the regulator optimally intervenes in the bank if its continuation withoutintervention would result in asset substitution that could magnify the eventual losses forcreditors and the DIF, which also give rise to social costs. Put differently, the optimalminimum capital requirement, below which the bank’s unsecured debt holders are bailedin, is determined by the level of capital below which the bank’s shareholders have a strongincentive to ‘gamble for resurrection’.

Since pγ < 1, a bank will always choose to reinvest in the safe asset if the macro stateis ‘good’ and the interim return is high:

p [γRH − iθ(1− e0)− (1− θ)] < RH − iθ(1− e0)− (1− θ) (2)

Hence, there is no need for a regulatory intervention in a ‘good’ macro state whenbanks receive a high return.

When the realised macro state is ‘bad’, some banks receiving very low returns mayhave the incentive to invest in the risky asset if their interim returns are low, in orderto ’gamble for resurrection’. This ex post moral hazard arises because shareholders arelimitedly liable, and interest rate on long-term debt cannot adjust once debt has beenissued: thus, a bank which has received a low interim asset return can increase expectedreturns for shareholders at the expense of debt holders by investing in risky, negativeNPV assets. More specifically, a bank has the incentive to gamble if the expected returnfrom investing the interim return in risky asset is higher than that of investing in the safeasset:

p[γRL − iθ(1− e0)− (1− θ)] < max {RL − iθ(1− e0)− (1− θ), 0} (3)

Rearranging the above, we obtain that banks will gamble for resurrection in the ab-sence of any regulatory intervention if RL < RT :

RT =1− p

1− pγ[(1− θ) + iθ(1− e0)] (4)

The investment in risky assets by a weakly capitalised bank ultimately magnifies ex-pected losses for its creditors and the DIF (and the associated social costs), thus creatinga rationale for an early intervention by the authorities to prevent asset substitution whenRL < RT . It can be shown that some of these banks are still solvent. By substitut-ing RL = RT from (4) into (1), we obtain the minimum equity ratio below which theauthorities need to intervene at t = 1 to prevent asset substitution by the bank (seeAnnex):

E∗ =p(γ − 1)

1− p> 0 (5)

We interpret E∗ as the optimal minimum regulatory capital requirement, or equiva-lently in our model, the trigger for resolution: if the equity ratio falls below this value,then the bank has the incentive to engage in socially sub-optimal asset substitution. Theincentive for undercapitalised banks to gamble for resurrection creates a case for earlyintervention by the authorities before the point of insolvency is reached.

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4.2 Bail in and resolution

We assume that, if the bank is found to be in breach of the minimum capital requirement(5) at t = 1, the resolution authority will use the following decision rule:

1. If there is sufficient amount of unsecured debt that can be converted into equity torestore E∗, then convert unsecured debt into equity and let the bank continue (‘bailin’).

2. If there is insufficient amount of unsecured debt that can be converted into equityto restore E∗, then wind down (or sell off) the bank. This requires liquidating thebank assets and compensating the claimholders in accordance with the hierarchy,i.e. the DIF is compensated first followed by the unsecured debt holders, and finallythe equity holders.

Importantly, we assume that resolution does not destroy the value of the bank, re-gardless of whether the bank is successfully recapitalised through bail in, or wound down.Instead, resolution only prevents the asset substitution by an undercapitalised bank, andthus benefits the uninsured creditors and the DIF relative to the counterfactual of no in-tervention. Thus, resolution is value preserving, in the sense that it maximises the sum ofthe expected returns to shareholders, unsecured debt holders, depositors and the DIF, al-though shareholders are prevented from engaging in a profitable risk-shifting opportunityand hence are made worse off.

As we clarify in Section 4.4, however, the imposition of losses on DIF and unsecureddebt holders gives rise to social costs that depend on the size of the losses imposed onthem. This implies that, regardless of whether the bank is recapitalised or wound down,the unsecured debt holders will suffer losses only if the bank is insolvent, i.e.:

RL < (1− θ) + iθ(1− e0) ≡ RS (6)

Thus, if the bank is solvent (i.e. RL ≥ RS), the unsecured debt holders receivetheir promised return iθ(1 − e0) in full. If the bank is insolvent, i.e. RL < RS,the unsecured debt holders will only receive the residual claim at wind down, given bymax {RL − (1− θ), 0}. The unsecured debt holders will receive nothing, although thedepositors are compensated in full by the deposit insurance full when RL < RD where.

RD ≡ 1− θ (7)

The stakeholder payoffs and the allocation of the asset return RL in the bad state aresummarised in Table 1:

4.3 The pricing of unsecured debt

The pricing of unsecured debt depends on the realisation of RL at the end of the periodt = 0. We assume that the bail in of unsecured debt is fully credible, such that the creditrisk is fully priced in the interest rate. Note that there are three ex post types of banksdepending on the realised return RL:

1. 0 ≤ RL ≤ RD (Type 1: insolvent in the low state, with zero recovery value forunsecured debt holders)

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Table 1: Payoffs of the claimholders in the bad state

RL ∈ (0, RD) RL ∈ (RD, RS) RL ∈ (RS , RT ) RL ∈ (RT , Rmax)

RA decision Resolve Resolve Resolve Don’t resolve

Stakeholder Payoffs

Shareholders 0 0RL − iθ(1− e0)−(1− θ)

RL − iθ(1− e0)−(1− θ)

GLAC holders 0RL − iθ(1− e0)−(1− θ) iθ(1− e0) iθ(1− e0)

Depositors (1− θ) (1− θ) (1− θ) (1− θ)DIF RL − (1− θ) 0 0 0

2. RD ≤ RL ≤ RS (Type 2: insolvent in the low state, with positive recovery value forunsecured debt holders)

3. RS ≤ RL ≤ Rmax (Type 3: solvent in the low state)

Type 1 banks will be insolvent in the low state, and the return RL will be insufficientto cover the liability to depositors. Thus, the unsecured debt holders of Type 1 banksare paid their promised return i if R = RH which occurs with probability q and receive0 with probability 1 − q. The unsecured debt holders need to earn the expected returnwhich is at least equal to the interest rate on the outside option of storage. Thus, theequilibrium interest rate for Type 1 banks’ unsecured debt, denoted as i1, is determinedby the following equilibrium condition:

qi1θ(1− e0) + (1− q)0 = θ(1− e0) (8)

This yields the following interest rate which includes a risk premium to reflect the factthat unsecured debt holders only get paid their promised i1θ(1− e0) with probability q.

i1 =1

q> 1 (9)

Thus, the expected profit of Type 1 banks at the end of period t = 0 (net of equityholders’ opportunity cost of investing in the bank) is given by:

Π1 = q(RH − (1− θ)− 1

qθ(1− e0))− δθe0

= q [RH − (1− θ)]− θ(1− e0)− δθe0 (10)

Note that the profits of Type 1 banks are distorted by the implicit subsidy from theDIF, as losses will be borne by the DIF in the bad state. It can be shown that ∂Π1

∂e0< 0

and ∂Π1

∂θ< 0 as long as δ > 1, so Type 1 bank will issue no more equity and unsecured

debt than required by the regulator at t = 1.Type 2 banks will also be insolvent in the low state, but the low return will be sufficient

to cover the liability to depositors, and the unsecured debt holders will receive the residualclaim in the low state. Thus, the equilibrium unsecured debt interest rate for Type 2banks, denoted as i2, is determined by the following condition:

qi2θ(1− e0) + (1− q)(RL − (1− θ)) = θ(1− e0) (11)

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Thus, the interest rate on unsecured debt for Type 2 banks is given by:

i2 =1

q− 1− q

q

RL − (1− θ)θ(1− e0)

(12)

The expected profit of Type 2 banks is given by:

Π2 = q

(RH − (1− θ)−

(1

q− 1− q

q

RL − (1− θ)θ(1− e0)

)θ(1− e0)

)− δθe0

= qRH + (1− q)RL − (1− θe0)− δθe0 (13)

Note that, because all losses in the low state will be borne by equity and unsecureddebt holders who can fully price all risks, profits of Type 2 banks are not distorted bythe presence of an implicit subsidy from the DIF. Type 2 banks will also issue as muchequity and unsecured debt as required by the regulator, as ∂Π2

∂e0< 0 and ∂Π2

∂θ< 0.

Type 3 banks will be solvent in the low state, such that the low asset return RL willbe sufficiently high to pay both depositors D and the unsecured debt holders their initialinvestment G in full, if the unsecured debt carries a risk free rate. This implies that theunsecured debt of Type 3 banks carries a risk free rate, such that:

i3 = 1 (14)

Thus, the expected profit of Type 3 banks is given by:

Π3 = q(RH − (1− θ)− θ(1− e0)) + (1− q)(RL − (1− θ)− θ(1− e0))− δθe0

= qRH + (1− q)RL − (1− θe0)− δθe0 (15)

As with type 2 banks, type 3 banks’ profits are not distorted by the implicit subsidyfrom the DIF, as the losses are fully borne by equity holders. The expected profits aresimply the expected value of the asset minus the liabilities. Type 3 banks will also issueas much equity and unsecured debt as required by the regulator, as ∂Π3

∂e0< 0 and ∂Π3

∂θ< 0.

The above implies that the expected profit of a bank at the start of period t = 0 isgiven by:

EΠ =

∫ 1−θ

0

{q [RH − (1− θ)]− θ(1− e0)} f(RL)dRL

+

∫ Rmax

1−θ{qRH + (1− q)RL − (1− θe0)} f(RL)dRL − δθe0

4.4 Ex ante social welfare

The ex ante regulatory requirements, θ and e0, will determine the share of each type inthe banking sector when the bad macro state – or a system-wide stress – materialisesex post. Thus, we now consider how the regulatory requirements θ and e0 can be opti-mally set ex ante: these parameters pin down what regulators refer to as the ”regulatoryequity buffer ratio” (E0 − E∗ = θe0 − E∗), and the minimum TLAC ratio requirement(E∗ +G = E∗ + θ(1− e0)), when the optimal minimum capital requirement E∗ is deter-mined by (5). In order to pin down interior solutions for these regulatory requirements, we

12

make the following assumptions in writing the social welfare function which the regulatormaximises to set the optimal θ and e0:

Assumption 1 (loss-absorbing equity): Equity can absorb losses ex post withoutimposing a social cost.

Assumption 2 (ex post costly bail in): The imposition of losses on the holdersof unsecured debt is associated with contagion externalities (social cost) ψ(LG ,T ) thatare convex in the loss given default LG,T imposed on unsecured debt holders of Type Tbank in the bad state (where T ∈ {1, 2}), so that ψ(0) = 0, ψ(LG,T ) ≥ 0 ∀LG,T > 0,ψ′(LG,T ) > 0, ψ′′(LG,T ) > 0.

Assumption 3 (ex post costly bail out): The imposition of losses on the depositinsurance fund (DIF) are associated with a deadweight cost χ(LD,1) that is quadraticwith respect to the loss given default LD,1 imposed on the DIF at resolution (for Type 1banks), where χ(0) = 0, χ(LD,1) ≥ 0 ∀LD,1 > 0, χ′(LD,1) > 0, and χ′′(LD,1) > 0.

Assumption 4 (value-preserving resolution): The resolution of a failing bankwill not destroy the recovery value of its assets, RL, which will be distributed amongstthe claimholders according to hierarchy. The resolution is triggered whenever E < E∗.

Assumption 4 implies that all of the bank’s asset returns are captured by a combi-nation of its shareholders, the unsecured debt holders, and the DIF at resolution. Thisassumption is made purely for expositional simplicity. We expect that dropping thisassumption would simply increase the optimal capital buffer by increasing the cost of bailin and bail out. Assumptions 1-3 imply that the imposition of a given loss on unsecureddebt holders or the DIF is socially more costly than the imposition similarly sized loss onequity holders, and that the social cost of a bank failure is captured by the externalitiescaused by the bail in and bail out. As discussed in Section 3, the externalities from bail-in(Assumption 2) could arise from the fact that, unlike equity, debt needs to be rolled over.Thus, there is a risk that the imposition of losses on unsecured debt holders of one bankleads raises concerns about the solvency of other banks, and thus create difficulties forother banks to roll over their maturing unsecured debt. This possibility is of particularconcern when the market is already under stress and there is a generalised concern aboutthe stability of the banking system as a whole. We assume that the externality functionis convex, thus capturing the possibility that, while a small scale bail in is done relativelyeasily without causing contagion, large scale bail-in is more likely to create investor un-certainty across the banking sector. Assumption 3 implies that the imposition of losseson the DIF is also associated with a social cost. We interpret this as a deadweight costassociated with funding any deficit of the DIF via distortionary taxes. For complete-ness, however, we do consider in Section 4.5 the cases in which Assumptions 2 and 3 aredropped.

Given the assumptions above, the ex ante social welfare, denoted as W , that the regu-lator will maximise to set the optimal θ and e0, subject to resolution occurring wheneverthe equity ratio falls below E∗ (given by (5)) at t = 1, is given by the expected return oninvestment of ex ante identical banks, net of the expected social costs of bail in and bailout (see Annex for derivation):

13

W ≡ R− δsθe0

− (1− q)

(∫ RS(i=i3)

RD

ψ(LG,2)f(RL)dRL +

∫ RD

0

[ψ(LG,1) + χ(LD,1)] f(RL)dRL

)(16)

where

R ≡ qRH + (1− q)∫ Rmax

0

RLf(RL)dRL

δs ≡ δ − 1

LG,1 ≡ i1θ(1− e0) (17)

LG,2 ≡ i2θ(1− e0)− [RL − (1− θ)] (18)

LD,1 ≡ (1− θ)−RL (19)

and, using (6), (14), the point of insolvency can be expressed as:

RS(i = i3) = (1− θ) + θ(1− e0) = 1− θe0 (20)

Note that δs in (16), which is explicitly derived in the Annex, can be interpreted asthe social opportunity cost of equity funding, given by the difference between the privateopportunity cost of equity investment and the safe rate of return. This social opportunitycost arises directly from our assumption that the credit and equity markets are segmented,and that equity investors face a higher opportunity cost of investing than creditors. Thesocial opportunity cost of equity funding, δs, captures the implicit sacrifice in economicoutput caused by forcing investors put their funds in bank equity instead of allowing themto raise funds from creditors who face a lower opportunity cost and thereby freeing upthe equity funding for productive investment opportunity in other sectors. We emphasisethat the social cost of equity funding in our model is not distorted by the implicit subsidyfrom the deposit insurance fund, and that it is lower than the private cost of equityfunding, as noted by Admati and Hellwig (2015).

It is worth clarifying at this point what the social costs and benefits of increasing theinitial capital requirement (i.e. the minimum capital requirement plus the capital buffer)are in our model. In our model, increasing the initial capital requirement produces thesocial benefit of reducing both the probability of a bank failure (by lowering the pointof insolvency, (20)) and the cost of a bank failure (by lowering the losses imposed onunsecured debt holders, (17) and (18)), but this comes at the cost of reducing output by‘crowding out’ investments in other sectors that banks’ equity investors could have fundedinstead. Thus, our analytical framework is broadly similar to Brooke et al. (2015), butwe distinguish the social cost of equity funding from the private cost more clearly.

4.5 Determinants of optimal regulatory requirements

We now examine the key determinants of the socially optimal trio of regulatory require-ments – the minimum capital requirement E∗, the TLAC requirement (E∗+G = θ−E∗)and the capital buffer (E0 − E∗ = θe0 − E∗) – through numerical simulations of (5) and(16). To do this, we assume the following quadratic functional forms for the social costsof bail in and bail out:

14

ψ(LG ,T ) = λGLG,T + λGL2G,T

χ(LD,T ) = λDLD,T + λDL2D,T

The derivation of the social welfare function under these assumptions is provided inthe Annex. Under the baseline calibration, we assume the following: p = 0.3, γ = 1.14,RH = 2, Rmax = 1.045, δs = 0.0025, λG = 0.02, λG = 0.0256, λD = 0.023, λD =0.0089, q = 0.9. The baseline parameterisations are chosen purely to reproduce theminimum TLAC, minimum capital and capital buffer requirements as set out by the FSBand BCBS. As these parameters are not chosen based on empirical estimates, we areprimarily interested in the directional (rather than quantitative) change in the optimaltrio of regulatory requirements caused by the changes in the assumptions about the keyparameters. But we note that the baseline parameterisations reflect the view that thesocial cost of equity funding is small, and that the cost of bail-in is smaller than the costof bail-out as long as the size of loss at insolvency is relatively small.

The tables below summarise the results from numerical simulation. The third columnof Table 2 below shows the optimal regulatory requirements under the baseline calibration:in this scenario, we obtain an optimal minimum capital requirement of 6.0% (consistentwith Basel III end-point Tier 1 capital ratio), capital buffer of 5.0% (consistent with BaselIII end-point capital conservation buffer of 2.5% plus G-SIB buffer of 1-2.5%), and TLACof 18.0% (consistent with the full implementation of the FSB’s Principles in 2022).

But a key unknown parameter is the cost of bail-in, given that there is little historicalprecedents for orderly imposition of losses on private bond holders of large banks. In thefourth column of Table 2, we present the ’low bail in cost’ scenario, in which λG = 0.02:i.e. the bail-in cost is both lower and less convex than in the baseline. The optimalequity buffer falls in this case to 4%, while the optimal TLAC rises to 25%. This isbecause in this scenario bail in is cheap compared to bail out, and unsecured debt is aclose substitute to equity in its ability to absorb losses without creating large externalities.Conversely, a higher and more convex bail-in cost (λG = 0.03, shown in the fifth columnof Table 2) would imply a lower optimal TLAC and a higher capital buffer than thebaseline. Note that the minimum capital requirement is invariant to the cost of bail in,as this is determined by the need to ensure that the bank has enough ’skin in the game’to incentivise it to invest in the safe asset.

Table 2: Optimal regulation and sensitivity to bail-in costs

Expression Baseline Low bailin cost High bailin cost

Minimum Capi-tal Ratio

E∗ 6.0% 6.0% 6.0%

Subordinateddebt

θ(1− e0) 12.0% 15.0% 10.5%

TLAC θ − E∗ 18.0% 21.0% 16.5%Capital buffer θe0 − E∗ 5.0% 4.0% 5.6%Minimum TLAC+ Capital Buffer

θ 23.0% 25.0% 22.0%

Bailout costs also matter. In the ’low bailout cost’ scenario presented in column 4

15

of Table 3 (in which λD = 0.008), both the capital buffer and TLAC are lower than thebaseline (shown in column 3). Conversely, in the ’high bailout cost’ scenario (shown incolumn 5) in which λD = 0.0095, so that bailout costs are both higher and more convex,both the capital buffer and TLAC can be considerably higher relative to the baseline.

Table 3: Optimal regulation and sensitivity to bail-out costs

Expression Baseline Low bailout cost High bailout cost

Minimum Capi-tal Ratio

E∗ 6.0% 6.0% 6.0%

Subordinateddebt

θ(1− e0) 12.0% 11.2% 12.6%

TLAC θ − E∗ 18.0% 17.2% 18.6%Capital buffer θe0 − E∗ 5.0% 3.8% 5.9%Minimum TLAC+ Capital Buffer

θ 23.0% 21.0% 24.5%

There are also other important parameters that determine the optimal regulatoryratios, as shown in Table 4. For example, a higher social cost of equity (δs = 0.00255)would imply that the regulator should require banks to hold a lower capital buffer thanthe baseline (column 3 in Table 4). Interestingly, stronger incentives for failing banks toengage in gambling, or asset substitution (γ = 1.15, shown in column 4 in Table 4) impliesthat the regulator should set a higher minimum capital requirement, but also a lowercapital buffer and a higher TLAC requirement, so as to maintain the sum of TLAC andcapital buffer (θ) the same as in the baseline: this is intuitive, as the cost of insolvency isunchanged from the baseline, while the need for early intervention has increased. Finally,if the probability of adverse macro shock becomes higher (q = 0.895, column 5 in Table 4),then the capital buffer should be increased while the TLAC requirement can be reduced.This reflects the higher expected social cost of bank failure, which calls for a higher capitaland lower unsecured debt to reduce the probability of bank failure. This analysis raisesan interesting possibility that, as the capital buffer is raised with increased risk of systemwide distress, the TLAC requirement may actually need to be reduced to allow banks tofund themselves with more capital and less debt.

For completeness, we also examine the extreme case in which both bail-out and bail-in are costless (λD = λG = 0 and λD = λG = 0): thus, our Assumptions 2 and 3

Table 4: Optimal regulation under different scenarios

Baseline High equity cost Strong gambling incentive Higher macro risk

Minimum Capi-tal Ratio

6.0% 6.0% 7.7% 6.0%

Subordinateddebt

12.0% 12.1% 12.0% 11.3%

TLAC 18.0% 18.1% 19.7% 17.3%Capital buffer 5.0% 3.4% 3.3% 8.7%Minimum TLAC+ Capital Buffer

23.0% 21.5% 23.0% 26.0%

16

are dropped, and both deposits and debt become perfect substitute to equity as loss-absorbing instruments. In this case, while a minimum capital requirement E∗ is stillneeded in order to prevent ex post moral hazard, and there is no need for a capital bufferin this case. A TLAC requirement, however, would still needed in the presence of ex-antemoral hazard, which we will consider in the next section, but its composition is irrelevantfor social welfare.

Finally, we examine what happens if only Assumption 2 is dropped. Suppose thatbail-in is costless (λG = λG = 0) but bail-out remains costly (as in the baseline). In thiscase, it is optimal to require banks to meet a TLAC requirement of 100% and a minimumcapital requirement of E∗ (=6% as in the baseline), but there will be no need for a capitalbuffer. This is intuitive: if bail-in is costless, then unsecured debt and equity are perfectsubstitutes ex post as loss absorbing instruments, and hence unsecured debt is favouredover equity as the latter is socially more costly ex ante. Deposits are socially less desirablethan unsecured debt in this case. We note, however, that deposits may carry benefitsthat are not considered in this model, such as the convenience of on-demand withdrawal,such that a 100% TLAC requirement may not be desirable even when bail-in is costless.

5 Ex-ante moral hazard and the impact of regulatory

policies

Thus far, we have assumed that the probability of a high return, q, is exogenous. Inthis section, we assume that q is determined endogenously by banks’ decisions over themonitoring of their projects, which requires them to exert costly effort. We then examinehow ex-ante regulation affects the incentive to monitor.

Suppose that, in the absence of any monitoring by banks at t = 0, the probability ofthe project yielding high return is qL. Following Holmstrom and Tirole (1997), supposethat, if banks choose to exert a monitoring effort which has private cost C, they canincrease the probability of project success from qL to qH , where qH > qL. We assume thatthe bank chooses the monitoring effort at the end of t = 0, after the realisation of RL

has been observed. At this point there is no information asymmetry between banks andpotential unsecured debt holders, so the market can fully price in the risk depending onwhether or not the bank monitors its investment at t=0.

5.1 Socially efficient monitoring

Monitoring is socially efficient for Type 1 banks with RL ∈ (0, RD), if the followingcondition holds:

qHRH + (1− qH) [RL − ψ(LG,1)− χ(LD,1)]− C ≥qLRH + (1− qL) [RL − ψ(LG,1)− χ(LD,1)] , ∀RL ∈ (0, RD)

In other words, the private monitoring cost C incurred by the banks must be suffi-ciently small relative to the social costs of bail in and bail out. This can be simplifiedto:

C ≤ C∗1 ≡ (qH − qL)(RH −RL) + (qH − qL)[ψ(LG,1) + χ(LD,1)], ∀RL ∈ (0, RD) (21)

The right hand side is the sum of the benefits from monitoring that accrue to banks’claimholders, (qH − qL)(RH −RL), and to the society in the form of reduced externalitiesfrom bail ins and bail outs, (qH − qL)[ψ(LG,1) + χ(LD,1)].

17

Similarly, monitoring is efficient for Type 2 banks with RL ∈ (RD, RS), as long as:

qHRH + (1− qH) [RL − ψ(LG,2)]−C ≥ qLRH + (1− qL) [RL − ψ(LG,2)] , ∀RL ∈ (RD, RS)

This can be reorganised as:

C ≤ C∗2 ≡ (qH − qL)(RH −RL) + (qH − qL)ψ(LG,2), ∀RL ∈ (RD, RS) (22)

Finally, monitoring is efficient for Type 3 banks with RL ∈ (RS, Rmax), as long as:

qHRH + (1− qH)RL − C ≥ qLRH + (1− qL)RL, ∀RL ∈ (RS, Rmax)

This can be reorganised as:

C ≤ C∗3 ≡ (qH − qL)(RH −RL), ∀RL ∈ (RS, Rmax) (23)

In what follows, we assume that C ≤ C∗1 , C ≤ C∗2 , and C ≤ C∗3 for all Types 1, 2, and3 banks, respectively. In other words, the regulator would always want to ensure thatbanks monitor their projects.

5.2 Private monitoring incentives

We now consider the monitoring incentives of a bank which seeks to maximise shareholderreturns in a context in which monitoring is socially desirable. From previous analysis inSection 4.3, we know that profits of Type 1 banks are distorted by the presence of implicitsubsidy from the DIF. Below, we demonstrate that only Type 1 banks are therefore proneto ex ante moral hazard; and that having a sufficiently high TLAC plus equity buffer, θ,can help reduce this.

For Type 1 banks withRL ∈ (0, RD), unsecured debt has zero recovery value ifR = RL.From (9), we know that the interest rate on unsecured debt is 1

qHif the bank chooses to

monitor, and 1qL

if it decides not to monitor, such that 1qH

< 1qL

. Type 1 banks willmonitor as long as the expected profit from monitoring, net of monitoring costs, exceedthe expected profit of not monitoring:

Π1(qH)− C > Π1(qL)

where Π1(.) is given by (10). Reorganising the above, we can show that Type 1 bankswill monitor their projects as long as θ ≥ θ∗where:

θ∗ =C

qH − qL−RH + 1 (24)

This shows that, unless θ is sufficiently high, Type 1 banks will be subject to moralhazard and will not monitor their project. Thus, the regulator can ensure that Type1 banks take actions to reduce risks by setting θ – TLAC plus equity buffer - above θ∗.We emphasise that the incentive to engage in moral hazard is influenced by the totalamount of losses that can be absorbed by private claimholders as opposed to taxpayers.In particular, the monitoring incentives are unaffected by the split of θ between equityand unsecured debt, as unsecured debt prices in the risks that banks are taking. Thus,

18

as long as θ is sufficiently high, distortions in monitoring effort arising from a limitedlyliable bank being funded by insured deposits is eliminated.3

For instance, for baseline parameterisations C = 0.45, qH = 0.9, qL = 0.5 and RH = 2,θ ≥ θ∗ =12.5% is needed to induce monitoring by Type 1 banks. But if there is anexogenous shock that reduces the probability of a high return that banks can achievethrough monitoring, θ will need to be raised in order to induce monitoring: for instance,for qH = 0.89, θ ≥ θ∗ =15.4% is needed to induce monitoring by Type 1 banks, consistentwith our previous analysis. Note that the optimal θ – which maximises social welfare(16) – could be higher than the minimum required to induce monitoring by Type 1 banks.

For Type 2 banks with RL ∈ (RD, RS) and Type 3 banks RL ∈ (RS, Rmax), we havealready seen that their profits are not distorted by the presence of the guarantee ondeposits, because the DIF will not have to pay out anything even in the bad scenario.Type 2 banks will monitor as long as:

Π2(qH)− C > Π2(qL)

where Π2(.) is given by (13). Similarly, Type 3 banks with RL ∈ (RS, Rmax) will monitoras long as:

Π3(qH)− C > Π3(qL)

where Π3(.) is given by (15). Reorganising the above, it can be shown that both Type 2and Type 3 banks will choose to monitor as long as:

C < (qH − qL)(RH −RL)

The right hand side is the benefits of monitoring that accrue to the banks’ claimholders.Comparing the above with (22) and (23), it is clear that Type 3 have the socially optimalincentive to monitor, regardless of the level of θ. This is because Type 3 banks willremain solvent even in the bad state, so that all costs and benefits are internalised. Bycontrast, Type 2 banks have a sub-optimal incentive to monitor, if (qH − qL)(RH −RL) <C < C∗2 : if so, Type 2 banks do not have the incentive to monitor, because the privatecost of monitoring outweighs the benefit, even though it is socially optimal for them tomonitor once the cost of bail in is taken into consideration. Note that Type 2 banks’incentive to monitor is independent of θ, because they will not impose losses on the DIFeven in the bad state and hence are not subject to moral hazard driven by the implicitsubsidy. This is why ex ante regulatory requirements cannot induce them to monitor, if(qH − qL)(RH −RL) < C < C∗2 .

6 Conclusion

A key contribution of our paper is to clarify what factors determine the optimal settingsof the trio of regulatory requirements: the minimum capital requirement, the TLACrequirement, the and capital buffer. We argue that these requirements should be setboth to eliminate the banks’ incentives to engage in ex ante and ex post moral hazard,and to minimise the expected social cost of a banking crisis.

3Or equivalently the distortion created from risk-insensitive interest rates on insured deposits is suffi-ciently small.

19

Our analysis illustrates that the optimal size of the capital buffer and TLAC, andthe optimal composition of TLAC, depend on the social cost of a crisis (i.e. the cost ofbail in and bail out), as well as the probability of a system-wide shock. This impliesthat the policymakers will need to take a view on how costly they expect bail-in – whichis yet to be tested – to be. If they expect system-wide externalities from bail in to belimited, then setting a low capital buffer and a high TLAC requirement would be optimal,when equity funding is socially expensive. By contrast, if policymakers fear that bail incould potentially cause contagion and hence could be costly, then a relatively low TLACrequirement and a high capital buffer would be desirable. Our analysis also raises aninteresting possibility that it may be desirable to make TLAC as well as capital buffertime-varying: in particular, as the risk of system-wide shock increases, the capital buffershould be raised and TLAC should be reduced in order to reduce the probability of bankfailure. This result could alternatively interpreted as stating that forbearance on theminimum TLAC requirement could be desirable in the same instances, and that it maynot give rise to the same adverse incentives associated with forbearance on the minimumcapital requirement.

References

[1] Acharya, V. V., Mehran, H., and Thakor, A. V. (2015). Caught between Scylla andCharybdis? Regulating bank leverage when there is rent seeking and risk shifting.Forthcoming in Review of Corporate Finance Studies.

[2] Admati, A. and Hellwig, M. (2015). The parade of the bankers’ new clothes continues:31 flawed claims debunked. Mimeo.

[3] Allen, F., Carletti, E., and Marquez, R. (2011). Credit market competition and capitalregulation. Review of Financial Studies, 24(4):983–1018.

[4] Allen, F., Carletti, E., and Marquez, R. (2015). Deposits and bank capital structure.Journal of Financial Economics, 118(3):601 – 619.

[5] Brooke, M., Bush, O., Edwards, R., Ellis, J., Francis, B., Harimohan, R., Neiss, K.,and Siegert, C. (2015). Measuring the macroeconomic costs and benefits of higher UKbank capital requirements. Financial Stability Working Paper, 35.

[6] De Nicolo, G., Gamba, A., and Lucchetta, M. (2014). Microprudential Regulation ina Dynamic Model of Banking. Review of Financial Studies, 27(7):2097–2138.

[7] Financial Stability Board (2015). Principles on loss-absorbing and recapitalisationcapacity of G-SIBs in resolution: Total loss-absorbing capacity (TLAC) term sheet.

[8] Hellmann, T. F., Murdock, K. C., and Stiglitz, J. E. (2000). Liberalization, MoralHazard in Banking, and Prudential Regulation: Are Capital Requirements Enough?American Economic Review, 90(1):147–165.

[9] Hoggarth, G. and Tanaka, M. (2006). Resolving banking crises: an analysis of policyoptions. Bank of England Working Paper, (293).

[10] Holmstrom, B. and Tirole, J. (1997). Financial Intermediation, Loanable Funds, andthe Real Sector. The Quarterly Journal of Economics, 112(3):663–91.

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[11] Mailath, G. J. and J.Mester, L. (1994). A Positive Analysis of Bank Closure. Journalof Financial Intermediation, 3(3):272–299.

[12] Martynova, N. and Perotti, E. (2015). Convertible bonds and bank risk-taking. DNBWorking Papers 480, Netherlands Central Bank, Research Department.

[13] Mendicino, C., Nikolov, K., and Suarez, J. (2016). Equity vs bail-in debt in banking:An agency perspective. Mimeo.

[14] Repullo, R. (2004). Capital requirements, market power, and risk-taking in banking.Journal of Financial Intermediation, 13(2):156–182.

[15] Vollmer, U. and Wiese, H. (2013). Minimum capital requirements, bank supervisionand special resolution schemes. Consequences for bank risk-taking. Journal of FinancialStability, 9(4):487–497.

A Technical Annex

A.1 Minimum Capital Requirement

The minimum capital requirement is obtained by substituting the gambling thresholdRL = RT from(4) into (1). Rearranging (1) we obtain RL = 1−θ+iθ(1−e0)

1−E∗ . Thus theminimum capital requirement E∗ solves the following condition:

1− θ + iθ(1− e0)

1− E∗=

1− p1− pγ

[(1− θ) + iθ(1− e0)] (25)

which becomes

1− E∗ =1− pγ1− p

Rearranging the above, we obtain (5).

A.2 Derivation of the social welfare function

To derive (16), note that the social welfare consists of two components: i) the expectedreturn from banks’ investment (which are divided between their claimholders) net of thesocial opportunity cost of funding that investment, and ii) the expected social cost ofbank failure. Under our assumption that credit and equity markets are segmented, thedepositors’ and creditors’ opportunity cost of providing the funding is equal to 1 (saferate of return) while the equity holders’ opportunity cost is δ. Thus, the first part of thesocial welfare function is given by:

Expected return on investment net of funding cost = R− (1− θ)− θ(1− e0)− δθe0

= R− 1− (δ − 1)θe0

= R− δsθe0 − 1 (26)

21

where R ≡ qRH +(1−q)∫ Rmax

0RLf(RL)dRL is the expected return on banks’ investment,

and δs ≡ δ − 1 is the social opportunity cost of funding the bank with equity instead ofdebt or deposits.

The second part of the social welfare function is given by the expected cost of bankfailure, which arises in a bad state when the bank becomes insolvent:

Expected social cost = −(1− q)(∫ RS(i=i3)

RD ψ(LG,2)f(RL)dRL

+∫ RD

0[ψ(LG,1) + χ(LD,1)] f(RL)dRL

)(27)

where losses imposed on uninsured debt holders and depositors of Type 1 and Type 2banks are given by:

LG,1 ≡ i1θ(1− e0)

LG,2 ≡ i2θ(1− e0)− [RL − (1− θ)]LD,1 ≡ (1− θ)−RL

and, using (6), (14), the point of insolvency can be expressed as:

RS(i = i3) = (1− θ) + θ(1− e0) = 1− θe0

Summing up (26) and (27) after dropping the constant −1 from (26), we obtain (16).

A.3 Social welfare function used for numerical simulation

We now derive the specific functional form of the welfare function (16), under the followingtwo assumptions.

ψ(LG ,T ) = λGLG,T + λGL2G,T

χ(LD,1) = λDLD,T

In this case, (16) can be rewritten in the following form:

W = R− (X + X)− (Y + Y )− δsθe0 (28)

where:

X = (1− q)λG∫ 1−θe0

1−θLG,2f (RL) dRL

X = (1− q)λG∫ 1−θe0

1−θL2G,2f (RL) dRL

Y = (1− q)∫ 1−θ

0

(λGLG,1 + λDLD,1) f (RL)

Y = (1− q)∫ 1−θ

0

(λGL

2G,1

)f (RL)

Expanding the above:

22

R = qRH + (1− q)∫ Rmax

0

RLf(RL)dRL

= qRH + (1− q)Rmax

2The losses suffered by unsecured debt holders of Type 1 and 2 banks in the event of

insolvency are given by:

LG,1 = i1θ(1− e0) =θ(1− e0)

q

LG,2 = i2θ(1− e0)− (RL − (1− θ))

=

(1

q− 1− q

q

RL − (1− θ)θ(1− e0)

)θ(1− e0)− (RL − (1− θ))

=1

q(1− θe0 −RL)

Substituting the above:

X =(1− q)λGqRmax

∫ 1−θe0

1−θ[(1− θe0 −RL)] dRL

=(1− q)λGqRmax

[[(1− θe0)RL]1−θe01−θ −

[R2L

2

]1−θe0

1−θ

]

=(1− q)λGqRmax

[(1− θe0) ((1− θe0)− (1− θ))− 1

2

[(1− θe0)2 − (1− θ)2

]]=

(1− q)λGqRmax

[θ (1− θe) (1− e0)− 1

2

[(1− θe0)2 − (1− θ)2

]]

X =(1− q)λGRmax

∫ 1−θe0

1−θ

(1

q(1− θe0 −RL)

)2

dRL

=(1− q)λGq2Rmax

∫ 1−θe0

1−θ

[(1− θe0)2 − 2RL(1− θe0) +R2

L

]dRL

=(1− q)λGq2Rmax

{(1− θe0)2[(1− θe0)− (1− θ)]− (1− θe0)[(1− θe0)

2 − (1− θ)2] +[(1− θe0)

3 − (1− θ)3]

3

}

=(1− q)λGq2Rmax

{θ(1− θe0)(1− θ)(e0 − 1) +

[(1− θe0)3 − (1− θ)3]

3

}

Y = (1− q)∫ 1−θ

0

[λGθ(1− e0)

q+ λD ((1− θ)−RL)

]f (RL) dRL

=(1− q)Rmax

[(λGθ(1− e0)

q+ λD(1− θ)

)RL

]1−θ

0

− (1− q)λDRmax

[R2L

2

]1−θ

0

=(1− q)Rmax

[(λGθ(1− e0)

q+ λD(1− θ)

)(1− θ)− λD

(1− θ)2

2

]=

(1− q)Rmax

[(λGθ(1− θ)(1− e0)

q

)+ λD

(1− θ)2

2

]23

Y =(1− q)Rmax

∫ 1−θ

0

[λG

(θ(1− e0)

q

)2]dRL

=(1− q)λGRmax

(θ(1− e0)

q

)2

(1− θ)

24